The vacuum is not empty. After removing all forms of matter and radiation, what remains could still carry a constant energy per unit volume. This constant can be ignored in all quantum calculations which care about energy differences, but must be considered in the context of gravity, which is sourced by all forms of energy. Albert Einstein’s equations of General Relativity imply that the vacuum energy density induces repulsive gravity, owing to its negative pressure. Hence, when the cosmic budget is dominated by the vacuum, the expansion of the Universe is expected to accelerate.
Four decades ago, the popular view among cosmologists and physicists was that the vacuum energy density is zero. Around the time that I arrived as junior faculty to the Harvard Astronomy department exactly thirty years ago — in January 1993, two graduate students of the department chair, Bob Kirshner, studied the light emitted by stellar explosions, the so-called supernovae. These were Brian Schmidt and Adam Riess. Both were awarded the Physics Nobel Prize in 2011 along with Saul Perlmutter for discovering that the expansion of the Universe is currently accelerating, based on measured distances to the cosmic light bulbs of Type Ia supernovae. Adam told me privately that he was inspired to pursue this path after attending my graduate class on Cosmology, in which I showed how to infer the cosmic acceleration from distance measurements. I was delighted to learn that at least students pay attention to what I say.
Thanks to this measurement, we now know the energy density of the vacuum — also called the cosmological constant or dark energy. It corresponds to an energy scale which is a billion times smaller than the mass of the electron, a milli-electron-volt. Can this measured value be explained from first principles?
From a fundamental perspective, the measured vacuum energy density is 123 factors of ten smaller than expected. Quantum mechanics predicts a zero-point vacuum energy of order the Planck scale, at which quantum gravity effects are important. The Planck scale is 31 orders of magnitude above the measured value of milli-electron-volt. The energy per unit volume scales as the 4th power of the energy scale, because the Heisenberg uncertainty principle implies an inverse relation between the distance scale probed at a given energy. As a result, the discrepancy in the vacuum energy density is 31 times 4 or 123 orders of magnitude. This constitutes the biggest blunder of fundamental physics over the past century.
It is therefore not surprising that when the first measurements of the cosmological constant were reported at a conference a quarter of a century ago, a string theorist told the observers that they should check their data because they must have made a mistake. In his mind, the cosmological constant should have been either 123 orders of magnitude larger or exactly zero. Back then, it was still hoped that string theory would explain why it is zero.
However, early attempts to explain the cosmological constant from fundamental principles failed. In 1987, the Nobel laureate Steven Weinberg wrote a paper suggesting that the fundamental “theory of everything” might admit many possible vacuum states that are realized in a region much larger than our observed spacetime, the so-called multiverse. Regions with much larger values of the cosmological constant would experience rapid expansion without sufficient time to make galaxies — and within them observers like us. Weinberg reasoned that our existence selects a spacetime region in which the cosmological constant would allow galaxies like the Milky Way to form and give birth to stars like the Sun and habitable planets like the Earth. This anthropic reasoning became popular among string theorists, who currently suggest that there are 10 to the power of 500 or perhaps even 10 to the power of 272,000 possible vacuum states.
But once again — data appears to contradict purely theoretical preferences. The deepest images from the Webb telescope show galaxies at redshifts above 15, when the matter in the Universe was denser than it is today by at least a factor of 4,000. The vacuum mass density is 2.3 times the matter density in the present-day Universe. This implies that even if the cosmological constant were thousands of times bigger than it is, galaxies with their stars and planets would have still formed in our Universe when it was merely a few percent of its current age. As I argued in a 2006 paper, this makes the anthropic argument for the cosmological constant less convincing. Could there be a better explanation for the measured value of the cosmological constant?
Any such explanation must combine quantum mechanics and gravity within a predictive theory of quantum gravity, which we do not possess. Nevertheless, a new paper I wrote with Mark Hertzberg, shows that under plausible assumptions, it is possible to relate the cosmological constant to known fundamental constants, such as Newton’s constant, Planck’s constant, the speed of light, as well as the mass and charge of the electron.
The seed for this paper originated as I was working last month on primordial black holes and realizing during a morning jog at sunrise a numerical coincidence. The quantum evaporation time of a black hole through Hawking radiation equals the current cosmic expansion time if the black-hole has a mass that provides a gravitational coupling with an electron that is equal in strength to the electric coupling between two electrons. This was a startling coincidence. Since the current cosmic expansion time is related to the vacuum energy density, I thought that this might offer a path towards explaining the measured value of the cosmological constant. After mentioning to Mark, he promptly came up with an insight on how to relate the two.
Based on plausible conjectures about quantum gravity in an exponentially accelerating universe, described by the vacuum-dominated spacetime derived by the Dutch physicist Willem de Sitter a century ago, our new paper related the value of the cosmological constant to parameters of the Standard Model of particle physics. In particular, we derived a relation between the cosmological constant and the electron’s mass and charge which happens to be of the order of the observed value. We also identified possible ways to test our proposal experimentally.
Perhaps the anthropic retreat to the unfalsifiable notion that we cannot predict the measured vacuum energy density from first principles was premature. In the words of Robert Frost, there might be a “Road Not Taken” with potentially low-hanging fruit that nobody had harvested before.
ABOUT THE AUTHOR
Avi Loeb is the head of the Galileo Project, founding director of Harvard University’s — Black Hole Initiative, director of the Institute for Theory and Computation at the Harvard-Smithsonian Center for Astrophysics, and the former chair of the astronomy department at Harvard University (2011–2020). He chairs the advisory board for the Breakthrough Starshot project, and is a former member of the President’s Council of Advisors on Science and Technology and a former chair of the Board on Physics and Astronomy of the National Academies. He is the bestselling author of “Extraterrestrial: The First Sign of Intelligent Life Beyond Earth” and a co-author of the textbook “Life in the Cosmos”, both published in 2021. His new book, titled “Interstellar”, is scheduled for publication in August 2023.
